Optimal. Leaf size=227 \[ -\frac {a^{10} A}{19 x^{19}}-\frac {a^9 (10 A b+a B)}{18 x^{18}}-\frac {5 a^8 b (9 A b+2 a B)}{17 x^{17}}-\frac {15 a^7 b^2 (8 A b+3 a B)}{16 x^{16}}-\frac {2 a^6 b^3 (7 A b+4 a B)}{x^{15}}-\frac {3 a^5 b^4 (6 A b+5 a B)}{x^{14}}-\frac {42 a^4 b^5 (5 A b+6 a B)}{13 x^{13}}-\frac {5 a^3 b^6 (4 A b+7 a B)}{2 x^{12}}-\frac {15 a^2 b^7 (3 A b+8 a B)}{11 x^{11}}-\frac {a b^8 (2 A b+9 a B)}{2 x^{10}}-\frac {b^9 (A b+10 a B)}{9 x^9}-\frac {b^{10} B}{8 x^8} \]
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Rubi [A]
time = 0.09, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77}
\begin {gather*} -\frac {a^{10} A}{19 x^{19}}-\frac {a^9 (a B+10 A b)}{18 x^{18}}-\frac {5 a^8 b (2 a B+9 A b)}{17 x^{17}}-\frac {15 a^7 b^2 (3 a B+8 A b)}{16 x^{16}}-\frac {2 a^6 b^3 (4 a B+7 A b)}{x^{15}}-\frac {3 a^5 b^4 (5 a B+6 A b)}{x^{14}}-\frac {42 a^4 b^5 (6 a B+5 A b)}{13 x^{13}}-\frac {5 a^3 b^6 (7 a B+4 A b)}{2 x^{12}}-\frac {15 a^2 b^7 (8 a B+3 A b)}{11 x^{11}}-\frac {b^9 (10 a B+A b)}{9 x^9}-\frac {a b^8 (9 a B+2 A b)}{2 x^{10}}-\frac {b^{10} B}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^{20}} \, dx &=\int \left (\frac {a^{10} A}{x^{20}}+\frac {a^9 (10 A b+a B)}{x^{19}}+\frac {5 a^8 b (9 A b+2 a B)}{x^{18}}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^{17}}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^{16}}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x^{15}}+\frac {42 a^4 b^5 (5 A b+6 a B)}{x^{14}}+\frac {30 a^3 b^6 (4 A b+7 a B)}{x^{13}}+\frac {15 a^2 b^7 (3 A b+8 a B)}{x^{12}}+\frac {5 a b^8 (2 A b+9 a B)}{x^{11}}+\frac {b^9 (A b+10 a B)}{x^{10}}+\frac {b^{10} B}{x^9}\right ) \, dx\\ &=-\frac {a^{10} A}{19 x^{19}}-\frac {a^9 (10 A b+a B)}{18 x^{18}}-\frac {5 a^8 b (9 A b+2 a B)}{17 x^{17}}-\frac {15 a^7 b^2 (8 A b+3 a B)}{16 x^{16}}-\frac {2 a^6 b^3 (7 A b+4 a B)}{x^{15}}-\frac {3 a^5 b^4 (6 A b+5 a B)}{x^{14}}-\frac {42 a^4 b^5 (5 A b+6 a B)}{13 x^{13}}-\frac {5 a^3 b^6 (4 A b+7 a B)}{2 x^{12}}-\frac {15 a^2 b^7 (3 A b+8 a B)}{11 x^{11}}-\frac {a b^8 (2 A b+9 a B)}{2 x^{10}}-\frac {b^9 (A b+10 a B)}{9 x^9}-\frac {b^{10} B}{8 x^8}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 220, normalized size = 0.97 \begin {gather*} -\frac {b^{10} (8 A+9 B x)}{72 x^9}-\frac {a b^9 (9 A+10 B x)}{9 x^{10}}-\frac {9 a^2 b^8 (10 A+11 B x)}{22 x^{11}}-\frac {10 a^3 b^7 (11 A+12 B x)}{11 x^{12}}-\frac {35 a^4 b^6 (12 A+13 B x)}{26 x^{13}}-\frac {18 a^5 b^5 (13 A+14 B x)}{13 x^{14}}-\frac {a^6 b^4 (14 A+15 B x)}{x^{15}}-\frac {a^7 b^3 (15 A+16 B x)}{2 x^{16}}-\frac {45 a^8 b^2 (16 A+17 B x)}{272 x^{17}}-\frac {5 a^9 b (17 A+18 B x)}{153 x^{18}}-\frac {a^{10} (18 A+19 B x)}{342 x^{19}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 208, normalized size = 0.92
method | result | size |
default | \(-\frac {a^{10} A}{19 x^{19}}-\frac {a^{9} \left (10 A b +B a \right )}{18 x^{18}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{17 x^{17}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{16 x^{16}}-\frac {2 a^{6} b^{3} \left (7 A b +4 B a \right )}{x^{15}}-\frac {3 a^{5} b^{4} \left (6 A b +5 B a \right )}{x^{14}}-\frac {42 a^{4} b^{5} \left (5 A b +6 B a \right )}{13 x^{13}}-\frac {5 a^{3} b^{6} \left (4 A b +7 B a \right )}{2 x^{12}}-\frac {15 a^{2} b^{7} \left (3 A b +8 B a \right )}{11 x^{11}}-\frac {a \,b^{8} \left (2 A b +9 B a \right )}{2 x^{10}}-\frac {b^{9} \left (A b +10 B a \right )}{9 x^{9}}-\frac {b^{10} B}{8 x^{8}}\) | \(208\) |
norman | \(\frac {-\frac {a^{10} A}{19}+\left (-\frac {5}{9} a^{9} b A -\frac {1}{18} a^{10} B \right ) x +\left (-\frac {45}{17} a^{8} b^{2} A -\frac {10}{17} a^{9} b B \right ) x^{2}+\left (-\frac {15}{2} a^{7} b^{3} A -\frac {45}{16} a^{8} b^{2} B \right ) x^{3}+\left (-14 a^{6} b^{4} A -8 a^{7} b^{3} B \right ) x^{4}+\left (-18 a^{5} b^{5} A -15 a^{6} b^{4} B \right ) x^{5}+\left (-\frac {210}{13} a^{4} b^{6} A -\frac {252}{13} a^{5} b^{5} B \right ) x^{6}+\left (-10 a^{3} b^{7} A -\frac {35}{2} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {45}{11} a^{2} b^{8} A -\frac {120}{11} a^{3} b^{7} B \right ) x^{8}+\left (-a \,b^{9} A -\frac {9}{2} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{9} b^{10} A -\frac {10}{9} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{8}}{x^{19}}\) | \(235\) |
risch | \(\frac {-\frac {a^{10} A}{19}+\left (-\frac {5}{9} a^{9} b A -\frac {1}{18} a^{10} B \right ) x +\left (-\frac {45}{17} a^{8} b^{2} A -\frac {10}{17} a^{9} b B \right ) x^{2}+\left (-\frac {15}{2} a^{7} b^{3} A -\frac {45}{16} a^{8} b^{2} B \right ) x^{3}+\left (-14 a^{6} b^{4} A -8 a^{7} b^{3} B \right ) x^{4}+\left (-18 a^{5} b^{5} A -15 a^{6} b^{4} B \right ) x^{5}+\left (-\frac {210}{13} a^{4} b^{6} A -\frac {252}{13} a^{5} b^{5} B \right ) x^{6}+\left (-10 a^{3} b^{7} A -\frac {35}{2} a^{4} b^{6} B \right ) x^{7}+\left (-\frac {45}{11} a^{2} b^{8} A -\frac {120}{11} a^{3} b^{7} B \right ) x^{8}+\left (-a \,b^{9} A -\frac {9}{2} a^{2} b^{8} B \right ) x^{9}+\left (-\frac {1}{9} b^{10} A -\frac {10}{9} a \,b^{9} B \right ) x^{10}-\frac {b^{10} B \,x^{11}}{8}}{x^{19}}\) | \(235\) |
gosper | \(-\frac {831402 b^{10} B \,x^{11}+739024 A \,b^{10} x^{10}+7390240 B a \,b^{9} x^{10}+6651216 a A \,b^{9} x^{9}+29930472 B \,a^{2} b^{8} x^{9}+27209520 a^{2} A \,b^{8} x^{8}+72558720 B \,a^{3} b^{7} x^{8}+66512160 a^{3} A \,b^{7} x^{7}+116396280 B \,a^{4} b^{6} x^{7}+107442720 a^{4} A \,b^{6} x^{6}+128931264 B \,a^{5} b^{5} x^{6}+119721888 a^{5} A \,b^{5} x^{5}+99768240 B \,a^{6} b^{4} x^{5}+93117024 a^{6} A \,b^{4} x^{4}+53209728 B \,a^{7} b^{3} x^{4}+49884120 a^{7} A \,b^{3} x^{3}+18706545 B \,a^{8} b^{2} x^{3}+17606160 a^{8} A \,b^{2} x^{2}+3912480 B \,a^{9} b \,x^{2}+3695120 a^{9} A b x +369512 a^{10} B x +350064 a^{10} A}{6651216 x^{19}}\) | \(244\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 243, normalized size = 1.07 \begin {gather*} -\frac {831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.37, size = 243, normalized size = 1.07 \begin {gather*} -\frac {831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.07, size = 243, normalized size = 1.07 \begin {gather*} -\frac {831402 \, B b^{10} x^{11} + 7390240 \, B a b^{9} x^{10} + 739024 \, A b^{10} x^{10} + 29930472 \, B a^{2} b^{8} x^{9} + 6651216 \, A a b^{9} x^{9} + 72558720 \, B a^{3} b^{7} x^{8} + 27209520 \, A a^{2} b^{8} x^{8} + 116396280 \, B a^{4} b^{6} x^{7} + 66512160 \, A a^{3} b^{7} x^{7} + 128931264 \, B a^{5} b^{5} x^{6} + 107442720 \, A a^{4} b^{6} x^{6} + 99768240 \, B a^{6} b^{4} x^{5} + 119721888 \, A a^{5} b^{5} x^{5} + 53209728 \, B a^{7} b^{3} x^{4} + 93117024 \, A a^{6} b^{4} x^{4} + 18706545 \, B a^{8} b^{2} x^{3} + 49884120 \, A a^{7} b^{3} x^{3} + 3912480 \, B a^{9} b x^{2} + 17606160 \, A a^{8} b^{2} x^{2} + 369512 \, B a^{10} x + 3695120 \, A a^{9} b x + 350064 \, A a^{10}}{6651216 \, x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 234, normalized size = 1.03 \begin {gather*} -\frac {x\,\left (\frac {B\,a^{10}}{18}+\frac {5\,A\,b\,a^9}{9}\right )+\frac {A\,a^{10}}{19}+x^9\,\left (\frac {9\,B\,a^2\,b^8}{2}+A\,a\,b^9\right )+x^2\,\left (\frac {10\,B\,a^9\,b}{17}+\frac {45\,A\,a^8\,b^2}{17}\right )+x^{10}\,\left (\frac {A\,b^{10}}{9}+\frac {10\,B\,a\,b^9}{9}\right )+x^4\,\left (8\,B\,a^7\,b^3+14\,A\,a^6\,b^4\right )+x^5\,\left (15\,B\,a^6\,b^4+18\,A\,a^5\,b^5\right )+x^7\,\left (\frac {35\,B\,a^4\,b^6}{2}+10\,A\,a^3\,b^7\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{16}+\frac {15\,A\,a^7\,b^3}{2}\right )+x^8\,\left (\frac {120\,B\,a^3\,b^7}{11}+\frac {45\,A\,a^2\,b^8}{11}\right )+x^6\,\left (\frac {252\,B\,a^5\,b^5}{13}+\frac {210\,A\,a^4\,b^6}{13}\right )+\frac {B\,b^{10}\,x^{11}}{8}}{x^{19}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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